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June 2008 The Willmore functional and instabilities in the Cahn-Hilliard equation
M. Burger, S.-Y. Chu, P. A. Markowich, C. -B Schonlieb
Commun. Math. Sci. 6(2): 309-329 (June 2008).


In this paper we are interested in the finite-time stability of transition solutions of the Cahn-Hilliard equation and its connection to the Willmore functional. We show that the Willmore functional locally decreases or increases in time in the linearly stable or unstable case respectively. This linear analysis explains the behavior near stationary solutions of the Cahn-Hilliard equation. We perform numerical examples in one and two dimensions and show that in the neighbourhood of transition solutions local instabilities occur in finite time. We also show convergence of solutions of the Cahn-Hilliard equation for arbitrary dimension to a stationary state by proving asymptotic decay of the Willmore functional in time.


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M. Burger. S.-Y. Chu. P. A. Markowich. C. -B Schonlieb. "The Willmore functional and instabilities in the Cahn-Hilliard equation." Commun. Math. Sci. 6 (2) 309 - 329, June 2008.


Published: June 2008
First available in Project Euclid: 1 July 2008

zbMATH: 1154.35009
MathSciNet: MR2433698

Primary: 35B35 , 35K57

Keywords: asymptotics , Cahn-Hilliard equation , stability , transition solutions , Willmore functional

Rights: Copyright © 2008 International Press of Boston

Vol.6 • No. 2 • June 2008
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